order	190
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₂×C₉×AGL(1,𝔽₅) (order 360) · Inn(G) = D₅ · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₂×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₁₉
unambiguous semidirect product	C₅⋊(C₂×C₁₉)

Splittings

Fitting, socle, Hall	C₅×C₁₉ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₁₉

	subgroup	G/H
center, hypercenter	C₁₉	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₁₉
Frattini	C₁	G
Fitting, socle	C₅×C₁₉	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₉

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 95

A minimal faithful permutation representation — 2 generators on 24 points:

(2,3)(4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24);(1,2,4,5,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(190, 1);